3 problems

Complementary norm conjecture. For $1\leq p\leq\infty$,

Inflection-point conjecture. There exists an inflection point $\chi_n(\alpha)\in\bigl(H_{\text{\scriptsize \boldmath }v_n}(\mathrm{e}^{-n}),\ln n\bigr)$ such that…

Let $N\geq 1$, let $D_N(z)=1+z+\dots+z^{N-1}$, and let $f(z)=c_0+c_1z^{n_1}+\dots+c_{N-1}z^{n_{N-1}}$ be an $N$-term trigonometric polynomial with $0<n_1<\dots<n_{N-1}$ and…